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Mathematics > Combinatorics

Abstract: This manuscript synthesizes almost fifteen years of research in algebraic
combinatorics, in order to highlight, theme by theme, its perspectives.
In part one, building on my thesis work, I use tools from commutative
algebra, and in particular from invariant theory, to study isomorphism problems
in combinatorics. I first consider algebras of graph invariants in relation
with Ulam's reconstruction conjecture, and then, more generally, the age
algebras of relational structures. This raises in return structural and
algorithmic problems in the invariant theory of permutation groups.
In part two, the leitmotiv is the quest for simple yet rich combinatorial
models to describe algebraic structures and their representations. This
includes the Hecke group algebras of Coxeter groups which I introduced and
which relate to the affine Hecke algebras, but also some finite dimensional Kac
algebras in relation with inclusions of factors, and the rational Steenrod
algebras. Beside being concrete and constructive, such combinatorial models
shed light on certain algebraic phenomena and can lead to elegant and
elementary proofs.
My favorite tool is computer exploration, and the algorithmic and effective
aspects play a major role in this manuscript. In particular, I describe the
international open source project *-Combinat which I founded back in 2000, and
whose mission is to provide an extensible toolbox for computer exploration in
algebraic combinatorics and to foster code sharing among researchers in this
area. I present specific challenges that the development of this project
raised, and the original algorithmic, design, and development model solutions I
was led to develop.